High-order Equivariant Flow Matching for Density Functional Theory Hamiltonian Prediction

TL;DR

QHFlow is a novel high-order equivariant flow matching framework that accurately predicts Hamiltonians conditioned on molecular geometry, significantly reducing errors and accelerating density functional theory calculations.

Contribution

It introduces a structured, equivariant flow-based model for Hamiltonian prediction, surpassing prior regression methods and improving DFT efficiency.

Findings

Reduces Hamiltonian error by 71% on MD17

Reduces Hamiltonian error by 53% on QH9

Speeds up DFT calculations by decreasing SCF iterations

Abstract

Density functional theory (DFT) is a fundamental method for simulating quantum chemical properties, but it remains expensive due to the iterative self-consistent field (SCF) process required to solve the Kohn-Sham equations. Recently, deep learning methods are gaining attention as a way to bypass this step by directly predicting the Hamiltonian. However, they rely on deterministic regression and do not consider the highly structured nature of Hamiltonians. In this work, we propose QHFlow, a high-order equivariant flow matching framework that generates Hamiltonian matrices conditioned on molecular geometry. Flow matching models continuous-time trajectories between simple priors and complex targets, learning the structured distributions over Hamiltonians instead of direct regression. To further incorporate symmetry, we use a neural architecture that predicts SE(3)-equivariant vector fields, improving accuracy and generalization across diverse geometries. To further enhance physical fidelity, we additionally introduce a fine-tuning scheme to align predicted orbital energies with the target. QHFlow achieves state-of-the-art performance, reducing Hamiltonian error by 71% on MD17 and 53% on QH9. Moreover, we further show that QHFlow accelerates the DFT process without trading off the solution quality when initializing SCF iterations with the predicted Hamiltonian, significantly reducing the number of iterations and runtime.